The Weak Repulsion Property

Transcription

1 The Weak Repulsion Property Alessandro Ferriero Chaire d Analyse Mathématique et Applications, EPFL, 115 Lausanne, Switzerland Abstract In 1926 M. Lavrentiev [7] proposed an example of a variational problem whose infimum over the Sobolev space W 1,p, for some values of p 1, is strictly lower than the infimum over W 1,. This energy gap is known since then as the Lavrentiev phenomenon. The aim of this paper is to provide a deeper insight into this phenomenon by shedding light on an unnoticed feature. Any energy that presents the Lavrentiev gap phenomenon is unbounded in any neighbourhood of any minimizer in W 1,p. We also show a finer result in case of regular minimizers and the repulsion property (observed by J. Ball and V. Mizel [3]) for any power α > 1 of a Lagrangian that exhibits the Lavrentiev gap phenomenon. Résumé En 1926 M. Lavrentiev [7] découvrit un exemple de problème variationnel dans lequel l infimum sur l espace de Sobolev W 1,p, pour des p 1, est strictement inférieur à l infimum sur l espace W 1,. Ce saut d énergie est connu sous le nom de phénomène de Lavrentiev. Le but de ce travail est de présenter une nouvelle caractéristique liée à ce phénomène qui permet de mieux le comprendre. Toutes les énergies qui manifestent le saut de Lavrentiev sont non bornées en tous les voisinages de chaque minimum en W 1,p. Finalement nous donnons un résultat plus précis concernant le cas où les minima sont réguliers, et nous démontrons la propriété de Répulsion (observée par J. Ball et V. Mizel [3]) pour toutes les puissances α > 1 d une Lagrangienne manifestant le phénomène de Lavrentiev. Key words: Lavrentiev phenomenon, repulsion property, singular phenomena, variational problems. MSC: 49J3, 49J45, 49N6. address: alessandro.ferriero@epfl.ch (Alessandro Ferriero). Preprint submitted to Elsevier 1 May 27

2 1. Introduction In 1926 M. Lavrentiev [7] published an example of an action functional I(u) := b a L(t, u, u)dt, whose infimum over the space W 1,p (a, b), for some p 1, is strictly lower than the infimum over the space W 1, (a, b), with fixed boundary conditions. This energy gap is known as the Lavrentiev phenomenon since then. It is a manifestation of the high sensibility of the variational formulation upon the set of admissible minimizers (considering that W 1, (a, b) is dense in W 1,p (a, b)). Notice that an unpleasant drawback of this phenomenon is the impossibility of computing the minimizer and the minimum of the energy by a standard finite-element scheme. One simple example exhibiting such energy gap is given by the Manià s action [8]: for any α 9/2, the functional I α (u) := 1 (u 3 t) 2 u α dt, with boundary conditions u() =, u(1) = 1, is such that { } { } inf I α : W 1,p t (, 1) < inf I α : W 1, t (, 1) (where W 1,p t (, 1) denotes the space t + W 1,p (, 1)), for p in [1, 3/2). More examples verifying this phenomenon have been given by several authors: see [5], [9] and references therein. A further singular phenomenon can be observed when α is strictly greater than 9/2. Any sequence {u n } in W 1, t (, 1) which converges almost everywhere in (, 1) to the minimizer of I α is such that I α (u n ) +. That is called repulsion property ([2]). It was observed by J. Ball and V. Mizel in 1985 [3] for a energy slightly different from I α. Fairly surprisingly, the closer we approximate the minimizer the farther we escape from the minimum value of the energy. The aim of this paper is to investigate the relations between Lavrentiev phenomenon and a weak repulsion property. Our main result states that, if I(u) := L(x, u, u)dx, (where is an open bounded set in R N, u is defined on with values in R m and L(x, u, w) is a Carathéodory function) presents the Lavrentiev phenomenon, then for any minimizer ū for I in W 1,p (), there exists a sequence {ū n} in W 1, () which converges to ū in W 1,p (), such that I(ū n ) + (where is a fixed function in W 1, () and W 1,p () denotes the space + W1,p ()). 2

3 In other words, in a variational model we observe the weak repulsion property as soon as we observe the Lavrentiev phenomenon (our main result is true under practically no assumptions). In dimension one and for a special but rather large class of Lagrangians, in [6] a similar result than the one we present here has been shown. Namely, the Lavrentiev phenomenon implies that any minimizer ū in W 1,1 is the limit of a sequence {ũ n } in W 1,1 such that I(ũ n ) is identically equal to + (by using the Fatou s lemma, from that result it is easy to deduce the existence of a sequence {ū n } in W 1, converging to ū in W 1,1 such that I(ū n ) +.) We would like to point out that the repulsion property is a property related to any sequence that converges to a minimizer whereas our weak repulsion property is a property related to (at least) one specific sequence among those that converge to a minimizer. On the contrary, our result would have been false since the Lavrentiev phenomenon does not imply the repulsion property (as the Manià s energy I α, for α = 9/2, shows). Our main theorem is therefore optimal in that sense. The paper is organized as follows. In section 2 we give the proofs of the afore mentioned claims on the Manià s functional for reader convenience. In section 3 we present our main result and the case of continuous and L q, q 1, minimizers. In section 4 we show the repulsion property for a modified Lagrangian. More precisely, we prove that, if an action with Lagrangian L exhibits the Lavrentiev phenomenon, then the action with the modified Lagrangian φ L manifests the repulsion property, for any function φ : R R with super-linear growth. 2. The Manià s Example For reader convenience, we present briefly an overview of the Manià s example. No original results are contained in this section (though the Manià s functional proposed here differs slightly from the usual one). An alternative source for this example can be found in [5], for instance. In Proposition 1, we show that, for α 9/2, the functional I α exhibits the Lavrentiev gap phenomenon and, in Proposition 2, that I α presents the repulsion property, for α > 9/2. We stress that the limit case α = 9/2 does not manifest the repulsion property. It is explained at the end of this section. Proposition 1 For any α 9/2, the functional I α (u) := 1 (u 3 t) 2 u α dt, with boundary conditions u() =, u(1) = 1, presents the Lavrentiev phenomenon, i.e. { } { } inf I α : W 1,p t (, 1) < inf I α : W 1, t (, 1), for p in [1, 3/2). PROOF. The Lagrangian associated to I α has non-negative values. Since I α evaluated in ū(t) = 3 t is identically zero, we obtain that ū is a minimizer of I α in W 1,p t (, 1), for 3

4 p in [1, 3/2). Let u be any function in W 1, t (, 1). Since u is essentially bounded, there exists a real number a in (, 1) such that u(t) < 3 t/2, for any t in [, a], and u(a) = 3 a/2. Hence, [ ] 2 t [u 3 t] 2 u α 2 3 t u α = t2 u α, for any t in [, a]. For α > 3, by the Hölder inequality, we obtain 3 a a a 2 = t 2/α t 2/α u dt t 2/(α 1) dt We conclude that, for any u in W 1, t (, 1), I α (u) for α 9/2. a = (u 3 t) 2 u α 3 2α/3 72 dt a 8 2 (α 1)/α a ( ) (α 1)/α α 1 a (α 3)/α α 3 ( ) (α 1) α 3 72 α t 2 u α dt a 1/α t 2 u α dt 1/α ( ) α 1 α 3 >, (1) α 1. Proposition 2 If α > 9/2, then I α (u n ) + for any sequence {u n } in W 1, t (, 1) converging a.e. to ū in (, 1). PROOF. For any natural number n, let a n in (, 1) be such that u n (t) < 3 t/2, for any t in (, a n ), and u(a n ) = 3 a n /2. By the convergence of u n to ū, we have that a n tends to. Using the inequality (1), since p is strictly greater then 9/2, we conclude that as n goes to. I α (u n ) a 3 2α/3 7 2 n 8 2 ( α 3 α 1 ) α 1 +, Notice that for α = 9/2, Proposition 2 is not true. Indeed, consider the sequence in (, 1) given by n 2/3 t, t [, 1/n], u n (t) := 3 t, t (1/n, 1], W 1, t 4

5 which converges a.e. to ū, and also in W 1,1 t (, 1) since 1 u n ū dt = 1/n n 2/3 3 1 t 2/3 dt = = 1/(3 3/2 n) (3 1 t 2/3 n 2/3 )dt + ( 1 4 ) n 1/3. 3 3/2 Let us verify that I α (u n ) is bounded for any α 9/2. In fact, for any n, 1/n 1/(3 3/2 n) 1/n 1/n ( 1 I α (u n ) = (n 2 t 3 t) 2 n 2α/3 dt = n 2α/3 (n 4 t 6 +t 2 +2n 2 t 4 )dt = n 2α/ ). 5 Therefore, for α = 9/2, I 9/2 (u n ) is bounded by the constant 1/7 + 1/3 + 2/5. Notice that the last equality also proves that, when α belongs to (, 9/2), I α does not manifest the Lavrentiev gap since I α (u n ) converges to = I α (ū). 3. The Weak Repulsion Property The framework we shall work with in the sequel is the following. Let be an open bounded set in R N. We denote by W 1,p (), p 1, the Sobolev space of vector-valued functions u : R m. Given in W 1, (), W 1,p () is as usual the space + W 1,p (). We assume that the Lagrangian L(x, u, w) : R m R m N R is a Carathéodory function, i.e. measurable with respect to x and continuous with respect to u and w, bounded from below by an integrable non-positive function ρ(x), uniformly in u, w. We deal with an energy I(u) := L(x, u, u)dx that presents the Lavrentiev phenomenon, i.e. for some p 1 inf{i(u) : u W 1,p ()} < inf{i(u) : u W1, ()}. Our main result is stated right below. (n 2/3 3 1 t 2/3 )dt Theorem 3 Let ū in W 1,p () be a minimizer of I. Then, there exists a sequence {ū n } in W 1, () which converges to ū in W1,p such that I(ū n ) +. PROOF. We prove the theorem for =. The general case can be reduced to that one replacing the Lagrangian L by L (x, u, w) := L(x, (x) + u, (x) + w). We can also suppose m = 1 since the general case can be obtained by proceeding componentwise as follows. 5

6 Fix ɛ >. Let {v n } be a sequence in C c () such that v n ū W 1,p () ɛ2 n and that v n and v n converge a.e. respectively to ū and ū. For any natural number n, consider the function L n in L 1 () given by L n (x) := max{l(x, v 1 (x), v 1 (x)),, L(x, v n (x), v n (x))}. STEP 1. We claim that [L n] + +, where [L n ] + denotes the positive part of L n. Suppose that this is not true. Notice that the sequence {L n } is non-decreasing and L n L(x, v n, v n ). Since by assumptions L(x, v n, v n ) ρ, we have also the inequality L(x, v n, v n ) max{[l n ] +, ρ}. (2) The monotonicity of the sequence {[L n ] + } implies that, for any measurable set E, [L n ] + l E < +. E By the Vitali-Hahn-Saks Theorem [1], we conclude that {[L n ] + } is equi-integrable and, by the estimate (2), {L(x, v n, v n )} is equi-integrable too. Since L(x, v n, v n ) converges a.e. to L(x, ū, ū), we have that L(x, v n, v n ) converges weakly- in L 1 () to L(x, ū, ū). That implies lim I(v n) = I(ū), n which contradicts the presence of the Lavrentiev phenomenon. Since [L n ] = min{l (x, v 1, v 1 ),, L (x, v n, v n )} ρ, we have that L n +. Set L := ρ and let E n be the measurable set defined by {x : L n (x) > L n 1 (x)}. Observe that L n = L(x, v n, v n ) on E n, whenever E n. By passing to a subsequence, we can suppose that E n, for any n. For k = 1,, n 1, set Fk n := E k \ (E k+1 E n ) and Fn n := E n. Notice that the are pairwise disjoint sets. We have that F n k L n = n L k χ F n = n L(x, v k k, v k )χ F n. (3) k k=1 k=1 STEP 2. We want to regularize the functions n k=1 v kχ F n k. Let C be the family of finite unions of disjoint open hypercubes of R d with faces parallel to the coordinate hyperplanes. The sets in the family C have piecewise smooth boundary. Since v n has compact support contained in, we can find a set n in the family C such that supp(v n ) n. Fix n in N. 6

7 For any δ > small enough, we define by induction on k, a set G n k (δ) in C such that n i=k F i n G n k (δ) except for a set of zero measure, Gn k (δ) Gn k 1 (δ) n (or G n k (δ) n, in case k = 1) and ( n ) Gn k(δ) \ Fi n δ. (4) i=k Let us denote by C(x n k (i); ln k (i)) the hypercubes in the family C with centre in xn k (i) and sides of length lk n(i) such that Gn k (δ) = m n k i=1 C(xn k (i); ln k (i)). Since G n k (δ) is an open set contained in n, for any η in (, ɛ) small enough, we can define the following set G n k (η) := m n k i=1 C(x n k(i); l n k (i) + η) n. For any k, there exists a function ρ n k (δ, η) in W1, (R n ) with values in [, 1] such that ρ n k (δ, η) is identically on Rn \ Gk n(η), ρ n k (δ, η) is identically 1 on Gn k (δ), ρ n k (δ, η) is normal a.e. to the boundary of Gn k (δ), ρ n k (δ, η) is identically equal to 2η 1 and, therefore, ρ n k (δ, η) L 1 (R n ) is bounded by a constant independent on k and n. We can therefore define a function u n (δ, η) in W 1, () by n u n (δ, η) := v 1 ρ n 1 (δ, η) + (v k v k 1 )ρ n k(δ, η). It follows directly from the definition that k=2 u n (δ, η) = v k, u n (δ, η) = v k on G n k(δ) \ n i=1,i k Gi n (η). STEP 3. We give the sought sequence {ū n }. By the construction of G n h (δ) and Gn h (η), we have that u n(δ, η) and u n (δ, η) converge a.e. respectively to v k and v k on Fk n, as δ and η tend to. Hence, the Fatou s lemma gives that L ndx lim η,δ I(u n (δ, η)). We can therefore chose δ n and η n such that L n dx 1 I(u n (δ n, η n )). (5) Moreover, recalling that { v n } is convergent in L p () and that ρ n k (δ n, η n ) is normal to G n k (δ n), δ n and η n can be chosen in such a way that we have as well the following inequalities v k v k 1 Lp ( G n k (δn)) η2 k 1, ( v k v k 1 ) ρ n k(δ n, η n ) 1/p L p () v k v k 1 L p ( G n k (δn)) + η2 7 k 1 (6)

8 (by passing to a subsequence of { v k } if needed). By the Poincaré inequality [4], there exists a constant c p > such that (v k v k 1 ) ρ n k(δ n, η n ) L p () v k v k 1 L p (G n k (ηn)) ρ n k(δ n, η n ) L (G n k (ηn))) m n k c p N 1/(2p) 2 v k v k 1 Lp (C j) j=1 (7) m n k c p (2N 1/2 η n ) 1/p 2 ( v k v k 1 ) ρ n k(δ n, η n ) 1/p Lp (C j) j=1 c p (2N 1/2 η n ) 1/p 8 ( v k v k 1 ) ρ n k(η n ) 1/p L p (), where {C j } mn k j=1 is the maximal subfamily of {C(xn k (i); ln k (i)+η n)} mn k i=1 such that ρn k (δ n, η n ) is not identically zero on any hypercube C j. Denote the functions u n (δ n, η n ) by u n and by c the constant c p (2N 1/2 ) 1/p By inequality (5), we have I(u n ) +. Furthermore, one can verify that, for any n, u n ū L p () ɛ and by (6) and (7), u n ū Lp () 2ɛc. (Recall that ɛ is an arbitrary positive constant given at the beginning of the proof.) Set ɛ n := n 1. For any n, the sequence {u n } admits a subsequence {u kn } such that u kn ū W 1,p () 3ɛ n c. Denoting the functions u kn by ū n, we have that the sequence {ū n } in W 1, () is such that and This concludes the proof. ū n ū W 1,p () I(ū n ) +. A local version of Theorem 3 can be briefly stated as follows: lim inf u 1,p ū, u W 1, I(u) I(ū) lim sup I(u) = +. () u 1,p () ū, u W 1, We do not enter into details since it follows directly from the proof. As a consequence of Theorem 3, we obtain the W 1,p local unboundedness of the energy I. We have therefore the following corollary. Corollary 4 Let ū in W 1,p () be a minimizer of I. If I is bounded in a neighbourhood of ū in W 1,p, then I does not manifest the Lavrentiev phenomenon. In case we assume the continuity of the minimizer ū in Theorem 3, we can improve the convergence of the repulsion sequence {ū n } to ū. More precisely: 8

9 Theorem 5 Let ū in C() W 1,p () be a minimizer of I. Then, there exists a sequence {ū n } in W 1, () which converges to ū in C W1,p such that I(ū n ) +. PROOF. The proof is slightly simpler than one of Theorem 3. We give a scheme of it using that theorem as reference. We prove the theorem for =. The general case can be reduced to that one replacing the Lagrangian L by L (x, u, w) := L(x, (x) + u, (x) + w). We can also suppose m = 1 since the general case can be obtained by proceeding componentwise as follows. Fix ɛ >. Let {v n } be a sequence in C c () such that v n ū W 1,p () ɛ2 n, v n ū C() ɛ2 n and that v n and v n converge a.e. respectively to ū and ū. For any natural number n, consider the function L n in L 1 () given by L n (x) := max{l(x, v 1 (x), v 1 (x)),, L(x, v n (x), v n (x))}. STEP 1 - STEP 2. Those are exactly the same as in Theorem 3. STEP 3. We give the sought sequence {ū n }. By the construction of G n h (δ) and Gn h (ɛ), u n(δ, ɛ) and u n (δ, ɛ) converge a.e. respec- to v k and v k on Fk n, as δ and ɛ tend to. Hence, the Fatou s lemma gives that tively L ndx lim δ I(u n (δ)). We can therefore chose δ n and ɛ n such that L n dx 1 I(u n (δ n, ɛ n )). (8) Denoting the function u n (δ n, ɛ n ) by u n, by the inequality (8), we have that I(u n ) +. Furthermore, one can verify that, for any n, u n ū C() ɛ and, by the Hölder s inequality, u n ū Lp () 2ɛ. (Recall that ɛ is an arbitrary positive constant given at the beginning of the proof.) Set ɛ n := n 1. For any n, the sequence {u n } admits a subsequence {u kn } such that u kn ū C() ɛ n and u kn ū W 1,p () 3ɛ n. Denoting the functions u kn by ū n, we have that the sequence {ū n } in W 1, () is such that and This concludes the proof. ū n ū C(), ū n ū W 1,p () I(ū n ) +. 9

10 In case the minimizer ū belongs to L q () with q 1 (but q ), we can be prove that: Theorem 6 Let ū in L q () W 1,p () be a minimizer of I. Then, there exists a sequence {ū n } in W 1, () which converges to ū in Lq W 1,p such that I(ū n ) +. PROOF. The proof is exactly the same as the one of Theorem 3 except for the initial approximating sequence {v n } C c () which is chosen so that v n ū W 1,p () ɛ2 n, v n ū L q () ɛ2 n and that v n and v n converge a.e. respectively to ū and ū. The reader might think that the weak repulsion property is satisfied by too many functionals. We would like to present below some simple examples which give proof of the contrary as we look for unbounded energies. Notice that the deal of our main result is to provide a test to exclude the presence of the Lavrentiev phenomenon. As we deal with solutions that we already know belong to W 1, (), it does not make any sense to apply Corollary 4 and Theorem 3 (that cannot even be applied in this case). Let us therefore restrict ourself to consider functionals with solutions in W 1,1 () \ W1, (). Let g be a function in [ 1 p<2 L p (, 1)]\L 2 (, 1), 1 g =, (for instance, define g(x) := x 1/2 1 x 1/2 dx) and consider the functional Denoting ū(t) := t I(u) := 1 [u (x) g(x)] 2 dx, for u in W 1,1 (, 1). g, we have that =, u = ū I(u) >, u W 1,1 = +, u W 1,2 (, 1) (, 1) \ W1,2 (, 1) Analogously, let f be a function in L 1 (, 1)\[ p>1 L p (, 1)], 1 f =, (for instance, define f(x) := [x log 2 (x/2)] 1 1 [x log2 (x/2)] 1 dx) and consider, for α > 1, the functional Denoting ũ(t) := t J α (u) := 1 [u (x) f(x)] α dx, for u in W 1,1 (, 1). f, we have that =, u = ũ J α (u) >, u W 1,1 (, 1) \ W1,α (, 1) = +, u W 1,α (, 1) 1..

11 The energies I and J α are indeed unbounded in any neighbourhood of ū and ũ in W 1,1 (, 1) and in fact we are in presence of the Lavrentiev phenomenon, and even of the repulsion property, as one can verify. 4. The Repulsion Property The Manià s functional for α = 9/2 shows that the Lavrentiev phenomenon does not implies the repulsion property. Nevertheless the implication holds true for any modified Lagrangian φ L, for any function φ from R to R with super-linear growth, i.e. φ(r)/r goes to +, as r tends to +. As particular case, consider φ(r) = r α, with α > 1. More precisely: (we recall that the we are in the same framework as the one described at the beginning of section 3) Theorem 7 Let ū in W 1,1 () be a minimizer of I. Then, φ L(x, u n, u n ) + for any sequence {ū n } in W 1, () such that u n and u n converge a.e. respectively to u and u. PROOF. The proof is elementary. Suppose that there exists a sequence {u n } in W 1, () such that u n and u n converge a.e. respectively to u and u and such that φ L(x, u n, u n ) l < +. From the lower bound on the Lagrangian, it follows that φ L (x, u n, u n ) is bounded. By the De la Vallée Poussin Theorem [1], we have the weak- compactness in L 1 () of {L(x, u n, u n )}, i.e. there exists a function L in L 1 () that is the weak- limit of a subsequence of {L(x, u n, u n )}. On the other hand, by the Carathéodory assumption on L, we know that L(x, u n, u n ) converges a.e. to L(x, ū, ū). Hence, L(x, u n, u n ) = L and I(u n ) I(ū), that contradicts the presence of the Lavrentiev gap phenomenon. Observe that in Theorem 7 it is not required any strong convergence on {u n }. The result is true for a.e. convergence. Corollary 8 If there exists a function φ with super-linear growth, increasing for r >, such that the functional φ 1 [L(x, u, u) ρ]dx 11

12 presents the Lavrentiev phenomenon, then I manifests the repulsion property, i.e. I(u n ) = L(x, u n, u n )dx + for any sequence {ū n } in W 1, () such that u n and u n converge a.e. respectively to u and u. PROOF. By Theorem 7 applied to the functional φ 1 [L(x, u, u) ρ]dx, we obtain that the energy [L(x, u, u) ρ]dx presents the repulsion property. The result therefore follows from the equality I(u) = [L(x, u, u) ρ]dx + ρdx. References [1] Ambrosio, L., Fusco, N., Pallara, D., Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2. [2] Ball, J.M., Singularities and computation of minimizers for variational problems, Foundations of Computational Mathematics, R. DeVore, A. Iserles and E. Suli Eds., London Mathematical Society Lecture Note Series, 284, pp 1 2. Cambridge University Press, 21. [3] Ball, J.M., Mizel, V.J., One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rational Mech. Anal. 9 (1985), no. 4, [4] Brezis, H., Analyse fonctionelle - Théorie et applications, Masson, Paris,1983. [5] Buttazzo, G., Giaquinta, M., Hildebrandt, S., One-dimensional variational problems. An introduction, Oxford Lecture Series in Mathematics and its Applications, 15. The Clarendon Press, Oxford University Press, New York, [6] Ferriero, A., The approximation of higher-order integrals of the calculus of variations and the Lavrentiev phenomenon, SIAM J. Control Optim. 44 (25), no. 1, [7] Lavrentiev, M., Sur quelques problemes du calcul des variations, An. Mat. Pura Appl., 4 (1926), pp [8] Manià, B., Sopra un esempio di Lavrentieff, Boll. Un. Mat. Ital., 13 (1934), pp [9] Mizel, V.J., Recent progress on the Lavrentiev phenomenon with applications, Differential equations and control theory (Athens, OH, 2), , Lecture Notes in Pure and Appl. Math., 225, Dekker, New York,